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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Baire category and $ B\sb r$-spaces


Author: Dominikus Noll
Journal: Proc. Amer. Math. Soc. 101 (1987), 671-678
MSC: Primary 54C10; Secondary 46A30
DOI: https://doi.org/10.1090/S0002-9939-1987-0911031-0
MathSciNet review: 911031
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Abstract: A topological space satisfying the open mapping theorem is called a $ {B_r}$-space. We investigate the question whether completely regular $ {B_r}$-spaces must be Baire spaces. The answer we obtain is twofold and surprising. On the one hand there exist first category completely regular $ {B_r}$-spaces. Examples are provided in the class of Lindelöf $ P$-spaces. On the other hand, we obtain a partial positive answer to our question. We prove that every suborderable metrizable $ {B_r}$-space is in fact a Baire space. We conjecture that this is true for metrizable $ {B_r}$-spaces in general. Our paper is completed by some applications. For instance, we establish the existence of a metrizable $ {B_r}$-space $ E$ whose square $ E \times E$ is no longer a $ {B_r}$-space.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0911031-0
Article copyright: © Copyright 1987 American Mathematical Society